Abstract:
The development of efficient reduced order models (ROMs) from a deep learning perspective enables users to overcome the limitations of traditional approaches [1, 2]. One drawback of the techniques based on convolutional autoencoders is the lack of geometrical information when dealing with complex domains defined on unstructured meshes.The present work proposes a framework for nonlinear model order reduction based on Graph Convolutional Autoencoders (GCA) to exploit emergent patterns in different physical problems, including those showing bifurcating behavior, high-dimensional parameter space, slow Kolmogorov-decay, and varying domains [3]. Our methodology extracts the latent space’s evolution while introducing geometric priors, possibly alleviating the learning process through up- and down-sampling operations. Among the advantages, we highlight the high generalizability in the low-data regime and the great speedup.Moreover, we will present a novel graph feedforward network (GFN), extending the GCA approach to exploit multifidelity data, leveraging graph-adaptive weights, enabling large savings, and providing computable error bounds for the predictions [4].This way, we overcome the limitations of the up- and down-sampling procedures by building a resolution-invariant GFN-ROM strategy capable of training and testing on different mesh sizes, resulting in a more lightweight and flexible architecture.References[1] Lee, K. and Carlberg, K.T. (2020) ‘Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders’, Journal of Computational Physics, 404, p. 108973. Available at: https://doi.org/10.1016/j.jcp.2019.108973.[2] Fresca, S., Dede’, L. and Manzoni, A. (2021) ‘A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs’, Journal of Scientific Computing, 87(2), p. 61. Available at: https://doi.org/10.1007/s10915-021-01462-7.[3] Pichi, F., Moya, B. and Hesthaven, J.S. (2024) ‘A graph convolutional autoencoder approach to model order reduction for parametrized PDEs’, Journal of Computational Physics, 501, p. 112762. Available at: https://doi.org/10.1016/j.jcp.2024.112762.[4] Morrison, O.M., Pichi, F. and Hesthaven, J.S. (2024) ‘GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications’, Computer Methods in Applied Mechanics and Engineering, 432, p. 117458. Available at: https://doi.org/10.1016/j.cma.2024.117458.

16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2

Universität Neuchatel

Neuchatel - St.Gallen - Zurich Seminar in Coding Theory and Cryptography

Prof. Dr. Jung Kyu Canci (Université de Neuchâtel &)

Arithmetic of Dynamical Systems Given by Rational Maps abstract

16:30 • Uni Neuchatel, B217

Universität Zürich

Seminar on Stochastic Processes

Zheng Fang (Universität Zürich, Switzerland)

Graduate Workshop Reinforcement

17:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H26

Thursday, December 12

Université de Genève

Physical Mathematics Seminar

Ideal Majtara (SISSA)

Partition function of gauge theories on the blowup and non-perturbative topological strings abstract

10:00 • Université de Genève, Conseil Général 7-9, Room 1-07

EPF Lausanne

Groups, Arithmetic and Algebraic Geometry Seminar

Charlie Reid (Mathematics MPIM, Leipzig)

Hitchin Components and Higher Complex Structures abstract

13:15 • EPF Lausanne, CM 1 517

ETH Zürich

[K-OS] Knot Online Seminar

Beibei Liu (Ohio State University)

Bounding the Dehn surgery number by 10/8 abstract

16:15 • online

Université de Genève

Colloque de Mathématiques

Tomasz Mrowka (MIT)

Forty years of Four Manifolds abstract

16:15 • Université de Genève, Conseil Général 7-9, Room 1-15

UniDistance Suisse

Mathematics Colloquium

Prof. Dr. Emre Mengi (Koc University)

Uniform Approximation of the Smallest Eigenvalue of a Large Parameter-Dependent Hermitian Matrix

17:00 • UniDistance Suisse, zoom

ETH Zürich

Kolloquium über Mathematik, Informatik und Unterricht

Sarah Zerbes (ETH Zürich)

Kongruente Zahlen: von Pythagoras zu Fermat

17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2

Friday, December 13

Universität Basel

Seminar in Numerical Analysis

Robert Scheichl (Heidelberg University)

Multigrid Monte Carlo Revisited: Fast Inference in High Dimensions abstract

Abstract:
The fast simulation of Gaussian random fields (GRFs) plays a pivotal role in many research areas such as uncertainty quantification, data science and spatial statistics. In theory, it is a well understood and solved problem, but in practice the efficiency and performance of traditional sampling procedures degenerates quickly when the random field is discretised on a grid with spatial resolution going to zero. Most existing algorithms, such as Cholesky factorisation samplers, do not scale well on large-scale parallel computers. On the other hand, stationary, iterative approaches such as the Gibbs sampler become extremely inefficient at high grid resolution. Already in the late 1980s, Goodman, Sokal and their collaborators wrote a series of papers aimed at accelerating the Gibbs sampler using multigrid ideas. The key observation is the intricate connection of random samplers, such as the Gibbs method, with iterative solvers for linear systems. They proposed the so-called multigrid Monte Carlo (MGMC) method - a random analogue of the multigrid method for solving discretised PDEs. We revisit the MGMC algorithm and provide rigorous theoretical justifications for the optimal scalability of the method for large scale problems, with a cost growing linearly with problem size. Most importantly we extend the method and the analysis to the Bayesian setting, i.e., GRFs conditioned on noisy data. By using bespoke, conditioned variants of the Gibbs sampler on each level of the multigrid hierarchy we are able to sample directly from the posterior with a fixed, grid-independent integrated autocorrelation time. Our theoretical analysis is confirmed by numerical experiments. We further generalise the approach by exploiting more flexible and robust grid hierarchies that were developed in the context of Algebraic Multigrid solvers. Finally, using existing PDE libraries, such as PETSs, the sampler is easily parallelised and scales optimally to large processor numbers. For further information about the seminar, please visit this webpage .

11:00 • Universität Basel, DMI, Spiegelgasse 5, 4051 Basel Seminarraum 05.001

EPF Lausanne

Talks in Actuarial Mathematics

Ilya Molchanov (University of Bern)

Geometric Gaussian Processes

14:00 • EPF Lausanne, Extranef 126

ETH Zürich

Number Theory Seminar

Dr. Xie Jianfeng (ETH Zurich)

The growth of Tate-Shafarevich groups in cyclic extensions abstract

14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43

EPF Lausanne

Statistics Seminar

Déborah Sulem (Universita della Svizzera Italiana)

Scalable and adaptive variational Bayes methods for Hawkes processes abstract

15:15 • EPF Lausanne, MA A3 30

ETH Zürich

Algebraic Geometry and Moduli Seminar

Dr. Pim Spelier (Utrecht University)

Gluing logarithmic curves: the logarithmic double ramification cycle and logarithmic Gromov-Witten invariants abstract

16:00 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43

address: Alessandra Naldi Windler, Departement Mathematik, HG G 51.4, ETH Zurich, Raemistrasse 101, CH-8092 Zurich
email: bulletin@math.ch | phone: +41 44 632 3684 | url: https://math.ch

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